internet economics כלכלת האינטרנט class 11 – externalities, cascades and the...

Internet Economicsכלכלת האינטרנט

Class 11 – Externalities, cascades and the Braess’s paradox.

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Today’s Outline

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• Network effects

• Positive externalities: Diffusion and cascades

• Negative externalities: Selfish routing.

Decisions in a network

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• When making decisions:– We often do not care about the whole population– Mainly care about friends and colleagues.

• E.g., technological gadgets, political views, clothes, choosing a job,. Etc.

What affects our decisions?

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• Possible reasons:– Informational effects:

Choices of others might indirectly point to something they know.“if my computer-geek friend buys a Mac, it is probably better than other computers”

– Network effects (direct benefit): My actual value from my decisions changes with the number of other persons that choose it.“if most of my friends use ICQ, I would be better off using it too”

Today’s topic

Main questions

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• How new behaviors spread from person to person in a social network.– Opinions, technology, etc.

• Why a new innovation fails although it has relative advantages over existing alternatives?

• What about the opposite case, where I tend to choose the opposite choice than my friends?

Network effects

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• My value from a product x is vi(nx): depends on the number nx of people that are using it.

• Positive externalities:– New technologies:

Fax, email, messenger, which social network to join, Skype.– vi(nx) increasing with nx.

• Negative externalities:– Traffic: I am worse off when more people use the same road as I.– Internet service provider: less Internet bandwidth when more people

use it.– vi(nx) decreasing with nx.

Network effects

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We will first consider a model with positive externalities.

Network effects

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• Examples:

VHS vs. Beta (80’s)

Internet Explorer vs. Netscape (90’s)

Blue ray vs. HD DVD (00’s)

Diffusion of new technology

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• What can go wrong?

• Homophily is a burden: people interact with people like themselves, and technologies tend to come from outside.– We will formalize this assertion.

• You will adapt a new technology only when a sufficient proportion of your friends (“neighbours” in the network) already adapted the technology.

A diffusion model

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• People have to possible choices: A or B– Facebook or mySpace, PC or Mac, right-wing or left-wing

• If two people are friends, they have an incentive to make the same choices.– Their payoff is actually higher…

• Consider the following case:– If both choose A, they gain a.– If both choose B, they gain b.– If choose different options, gain 0.

A BA (a,a) (0,0)B (0,0) (b,b)

A diffusion model (cont.)

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• So some of my friends choose A, some choose B. What should I do to maximize my payoff?

• Notations:– A fraction p of my friends choose A– A fraction (1-p) choose B.

• If I have d neighbours, then: – pd choose A – (1-p)d choose B.

• With more than 2 agents: My payoff increases by a with every friend of mine that choose A. Increases by b for friends that choose B.

Example:If I have 20 friends, and p=0.2:pd=4 choose A(1-p)d=16 choose BPayoff from A: 4aPayoff from B: 16b


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• Therefore:– Choosing A gain me pda– Choosing B will gain me (1-p)db

• A would be a better choice then B if:pda > (1-p)db

that is, (rearranging the terms)p > b/(a+b)

• Meaning: If at least a b/(a+b) fraction of my friends choose A, I will also choose A.

• Does it make sense? When a is large, I will adopt the new technology even when just a few of my friends are using it.


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• This starts a dynamic model:– At each period, each agent make a choice given the choices of his

friends.– After everyone update their choices, everyone update the choices

again,– And again,– And again,– …

• What is an equilibrium?– Obvious equilibria: everyone chooses A.

everyone chooses B.– Possible: equilibria where only part of the population

chooses A.

“complete cascade”

Diffusion

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• Question:Suppose that everyone is initially choosing B– Then, a set of “early adopters” choose A– Everyone behaves according to the model from previous slides.

• When the dynamic choice process will create a complete cascade?– If not, what caused the spread of A to stop?

• Answer will depend, of course, on:– Network structures– The parameters a,b– Choice of early adopters

B

B

BB

B

BB

B

B B

BB

B

A

A

A

Example

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• Let a=3b=2

• We saw that player will choose A if at leastb/(a+b) fraction of his neighbours adopt A.

• Here, threshold is 2/(3+2)=40%

Example 1

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Example 1

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Two early adopters of the technology A

Example 1

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Example 1

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A full cascade!

Example 2

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Let’s look at a different, larger network

Example 2

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Again, two early adopters

Example 2

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Example 2

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Example 2

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Dynamic process stops: a partial cascade

Partial diffusion

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• Partial diffusion happens in real life?

– Different dominant political views between adjacent communities.

– Different social-networking sites are dominated by different age groups and lifestyles.

– Certain industries heavily use Apple Macintosh computers despite the general prevalence of Windows.

Partial diffusion: can be fixed?

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• If A is a firm developing technology A, what can it do to dominate the market?– If possible, raise the quality of the technology A a bit.

• For example, if a=4 instead of a=3, then all nodes will eventually switch to A. (threshold will be lower)

Making the innovation slightly better, can have huge implications.

– Otherwise, carefully choose a small number of key users and convince them to switch to A.

• This have a cost of course, for example, giving products for free or invest in heavy marketing. (“viral marketing”)

• How to choose the key nodes?• (Example in the next slide.)

Example 2

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For example: Convincing nodes 13 to move to technology A will restart the diffusion process.

Cascades and Clusters

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• Why did the cascade stop?

• Intuition:the spread of a new technology can stop when facing a “densely-connected” community in the network.


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• What is a “densely-connected” community?If you belong to one, many of your friends also belong.

• Definition: a cluster of density p is a set of nodes such that each nodes has at least a p-fraction of her friends in the cluster.

h

A 2/3 cluster

h


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A 2/3 cluster


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• Note: not every two nodes in a cluster have much in common– For example:

• The whole network is always a p-cluster for every p.• Union of any p-clusters is a p-cluster.


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In this network, two 2/3-clusters that the new technology didn’t break into. Coincidence?


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• It turns out the clusters are the main obstacles for cascades.

• Theorem:Consider: a set of initial adopters of A, all other nodes have a threshold q (to adopt A).Then:

1. if the other nodes contain a cluster with greater density than 1-q, then there will be no complete cascade.

2. Moreover, if the initial adopters did not cause a cascade, the other nodes must contain a cluster with a density greater than 1-q.

Previously we saw a threshold q=b/(a+b)


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In our example, q=0.4 cannot break into p-clusters where p>0.6

Indeed: two clusters with p=2/3 remain with B.


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Let’s prove this

part.


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• Assume that we have a cluster with density of more than 1-q• Assume that there is a node v in this cluster that was the first

to adopt A• We will see that this cannot happen:

• Assume that v adopted A at time t.

• Therefore, at time t-1 at least q of his friends chose A

• Cannot happen, as more than 1-q of his friends are in the cluster• (v was the first one to

adopt A)


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Let’s prove this

part.


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• We now prove: not only that clusters are obstacles to cascades, they are the only obstacle!

• With a partial cascade: there is a cluster in the remaining network with density more than 1-q.

• Let S be the nodes that use B at the end of the process.

• A node w in S does not switch to A, therefore less than q of his friends choose A

The fraction of his friends that use B is more than 1-q

The fraction of w’s neighbours in S is more that 1-q

S is a cluster with density > 1-q.

Today’s Outline

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• Network effects

• Positive externalities: Diffusion and cascades

Negative externalities: Selfish routing.

Negative externalities

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• Let’s talk now about setting with negative externalities: I am worse off when more users make the same choices as I.

• Motivation: routing information-packets over the internet.– In the internet, each message is divided to small packets

which are delivered via possibly-different routes.

• In this class, however, we can think about transportation networks.

Example

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• Many cars try to minimize driving time.• All know the traffic congestion (גלגלצ, PDA’s)

Example

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• Negative externalities: my driving time increases as more drivers take the same route.

• Nash equilibrium: no driver wants to change his chosen route.

• Or alternatively:– Equilibrium: for each driver, all routes have the

same driving time.• (Otherwise the driver will switch to another route…)

Example

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• Our question: are equilibria efficient?– Would it be better for the society if someone told

each driver how to drive???

• We would like to compare:– The most efficient outcome (with no incentives)– The worst Nash equilibrium.

• We will call their ratio: price of anarchy.

Example

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• Efficient outcome: efficiency=4+4=8• (Worst) Nashe Equilibrium: efficiency=2+2=4

• Price of anarchy: 1/2

Cooperate Defect

Cooperate -1, -1 -5, 0Defect 0, -5 -3,-3

Cooperate Defect

Cooperate 4, 4 0, 5Defect 5, 0 2,2

Example 1

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• Efficient outcome: splitting traffic equally– expected cost: ½*1+1/2*1/2=3/4

• The only Nash equilibrium: everyone use lower edge.– Otherwise, if someone chooses upper link, the cost in the

lower link is less than 1.– Expected cost: 1*1=1

“Price of anarchy”: 3/4

C(x)=x

C(x)=1

• c(x) – the cost (driving time) to users when x users are using this road.

• Assume that a flow of 1 (million) users use this network.

S T

Example 2

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• In equilibrium: half of the traffic uses upper routehalf uses lower route.

• Expected cost: ½*(1/2+1)+1/2*(1+1/2)=1.5

c(x)=x

c(x)=1

S T

c(x)=x

c(x)=1

Example 3

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• The only equilibrium in this graph:everyone uses the svwt route.– Expected cost: 1+1=2

• Building new highways reduces social welfare!?

c(x)=x

c(x)=1

S T

v

W

c(x)=x

c(x)=1

c(x)=0

Now a new highway was constructed!

!!!!

Braess’s Paradox

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• This example is known as the Braess’s Paradox:

sometimes destroying roads can be beneficial for society.

c(x)=x

c(x)=1

S T

v

W

c(x)=x

c(x)=1

c(x)=0

Now a new highway was constructed!

Selfish routing, the general case

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• What can we say about the “price of anarchy” in such networks?

• We saw a very simple example where it is ¾

• Actually, this is the worst possible:

Theorem: when the cost functions are linear (c(x)=ax+b), then the price of anarchy in every network is at least ¾.

Summary

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• Network effects are important in many different aspects of the Internet.

• Explain many of the phenomena seen in the last couple of decade (and before…)

internet economics כלכלת האינטרנט class 11 – externalities, cascades and the...

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